Series and that old ivar error

Question

I am trying to compare plots of the function (1+x)^{1/3} and its power series. The code for the power series generates an error message referring to an invalid variable: 'General::ivar: -1.09993 is not a valid variable."' This message occurred with a minimum x of -1.1. Whatever value is given to the minimum x, the error message is always .00007 or .00008 greater. Yet it continues and plots the curve to maximum x. I would like to know the reason for the error message.

A previous similar question by @fasttouch "Plot Series output?" was closed as off-topic. The apparent correct answers were to apply Normal[], which I have done, and Evaluate[], which here seems to have no effect, as the curves with and without it coincide. If there is a trivial syntax error, it would not surprise me, but I can't see it, nor does the answer appear to be easily found in the documentation. I have Mathematica 8.

I am also wondering why I need to reference the series with an index in Plot, but not in Print.

x0 = 0; n = 30;

Print["Series up to 5th power: ", 
  Normal[Series[Power[1 + x, 1/3], {x, x0,  5}]]];

Plot[{
  Evaluate[Normal[Series[Power[1 + x, 1/3], {x, x0,  n}][[1]]]],
  Normal[Series[Power[1 + x, 1/3] + .01, {x, x0,  n}][[1]]]
  }, {x, -1.1, 2.5}, Background -> White]

After submitting this, I find that all works well if I write it this way:

s[x_] := Normal[Series[Power[1 + x, 1/3], {x, x0,  n}]];

Plot[Evaluate@s[x], {x, -1, 1}]

My current guess is that there was some confusion of the x's in Series[] and Plot[].

Answer 1

You have the x overloaded in too many places. You use it as free variable for the Series, then you use it for the plot command variable. I think this error happens because

 ?? Plot

show it has HoldAll. Then the x for the plot takes effect before the Series is expanded. But this 'x' is now a number, so Series complains as it needs a symbol.

But best to Keep things simple. One way:

ClearAll[x]
x0 = 0; n = 30;
ser = Normal[Series[Power[1 + x, 1/3], {x, x0, n}]];
f = (1 + x)^(1/3);
Plot[Evaluate@{ser, f}, {x, -1.1, 2.5}, Background -> White, 
     Frame -> True, PlotLegends -> {"Series", "function"}]